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$(\partial_t - \Delta) u = -u\cdot\nabla u - \nabla p$
$div(u)=0$
Here $u$ is a two or three dimensional vector field on $\mathbf{R}^2$ or $\mathbf{R}^3$.  This is fairly typical of partial differential equations in applied physics and engineering and Russians had a monumental project of these things to classify the solutions in some way because without the superposition principle one does not have any simple way of classifying solutions. Quantum mechanics with its linear equations are an exception that is probably just an approximation from what I can tell.  Now it’s not worthwhile challenging quantum mechanics because a replacement theory requires a lot more vital energy and will and resources than I will be able to command anytime soon but it’s a real hope.  My serious comment on this will be that it is very dangerous to go crazy mythologizing intepretation based on a linear approximation.  Physicists and scientists in general eschew, as a sacred tradition, anything that smells of mysticism.  It’s a religious dogma for them rather than anything else.  They are as adamant about this as any Catholic priest will be about his little God.  Well, I’m not a pure empiricist and I am as mystical sometimes as Emerson’s Over-Soul.  I think the univese is governed by magic beyond the Hell on Earth which came with nuclear bombs and believe that Churchill and Roosevelt should be burning in Hell for what they had brought to this planet, instruments that belong in the lowest circle of Dante’s Inferno.  This is not what science is about.  This has to change.